Isomorphism in Digital Elevation Models and Its Implication to Interpolation Functions

Terrain is an ordered surface where locations relate to each other through their elevations. Preservation of this topographic orderliness by an interpolation method so that “if point A is higher than point B in the terrain, the interpolated elevation of A remains higher” is important to assure the practical value of the resultant DEM. Based on the concepts in group theory, this paper points out that a DEM must be isomorphic to terrain surface in order to preserve the topographic orderliness. Such a DEM, if generated through interpolation, can only be obtained if the interpolation function is an isomorphism. Two necessary conditions for isomorphism are identified: a one-to-one relationship between the topographic surface and the surface corresponding to the interpolation function, and the feasibility to configure the topographic surface into monotonic patches during interpolation. The isomorphism of three widely used interpolation methods is examined. It is found that linear interpolation in 1D and TIN are both isomorphisms, meaning that a DEM interpolated by either method can preserve the topographic orderliness. In contrast, bilinear interpolation is not an isomorphism. Considering the practical challenges in assuring the necessary conditions of isomorphism in each method, linear interpolation in 1D is recommended as the optimal method to interpolate a DEM.